Preheat the oven to 350°F

Generously coat 2 rondeau pans with olive oil and set over high heat. Generously season the lamb shanks with salt and add to the pans. Brown well on all sides, then transfer to a baking sheet This is a very important step; do not rush it.

Meanwhile, in a food processor, combine the onion, carrots, celery and garlic  and puree until a coarse paste forms. Set aside. 

Discard the excess fat from one of the pans, then add a little more olive oil to coat the bottom. Add the pureed vegetables, season with salt and cook, stirring occasionally, until the vegetables are very brown and aromatic, about 20 minutes. The browned vegetables should form a crust on the bottom of the pan. Do not let this burn; it is where a lot of the flavor is. Do not rush this step as this is where you will develop the brown color and flavor.

Add the tomato paste and brown for 5 minutes. Stir in the wine, rosemary and thyme. Cook, stirring frequently, until the wine has reduced by about half.

Add the shanks to the pan and pour in 3 to 4 cups water. They should be submerged; if not, add more water. Add the bay leaves, cover and transfer to the oven. Cook until the meat is incredibly tender and falling off the bone, 2 1/2 to 3 hours, turning the shanks over about halfway through the cooking time. Check the shanks every 45 minutes or so. If the liquid has reduced too much, add more water. Skim off the fat as you go. To maximize browning, remove the lid during the last 30 minutes of cooking.

In a medium saucepan over medium heat, bring the water to a boil. Add the polenta and cornmeal in a slow, steady stream, whisking for 1 minute to prevent lumps from forming. Reduce the heat and simmer, stirring constantly with a wooden spoon. Add the pumpkin and continue stirring until the polenta begins to pull away from the sides of the pan, 15 to 20 minutes. Stir in the nutmeg,  cheese and butter. 

Serve the polenta immediately while hot and creamy with the braised lamb shanks.